Template problems in group theory

Mar 29, 2025 10:01 PM

Let G and H be groups of order 10 and 15, respectively. Prove that if there is a nontrivial homomorphism ϕ:GH, then G is abelian.


Let n be a number between 0 and 10. Compute n111(mod11), expressing your answer as a number between 0 and 10. Give as detailed a proof as you can, justifying every step, no matter who trivial you think it is.


Let Sn denote the symmetric group on n letters.

  1. Is the element (1234)(25346)(153247)S7 even or odd? Indicate your reasoning.
  2. Find the order of (134)(243)(134)S4. Show all work.
  3. Write (1523)(2134)(1523)1S5 in disjoint cycle form. Show all work.

Determine with proof the automorphism group Aut(V) of the Klein 4-group V={e,a,b,ab}. To what familiar group is it isomorphic?


Determine the number of group homomorphisms ϕ between the given groups. Here K4 denotes the Klein four-group (also known as Z/2Z×Z/2Z) and S3 denotes the symmetric group on three elements.

  1. ϕ:K4Z/2Z
  2. ϕ:Z/2ZK4
  3. ϕ:S3K4
  4. ϕ:K4S3

Without using Cauchy's Theorem or the Sylow theorems, prove that every group of order 21 contains an element of order three.


Explicitly list all group homomorphisms f:Z/6ZZ/12Z.


Let C be a (possibly infinite) cyclic group, and let Aut(C) and Inn(C) be the groups of automorphisms and inner automorphisms, respectively. (Recall an automorphism γ is inner if it is given by conjugation: γ(b)=aba1 for some aC.)

  1. Describe Aut(C) and Inn(C) in familiar terms, as groups you would study in a first algebra course. Prove your result. (Hint: Where do generators go?)
  2. Write Aut(Z12) down explicitly, giving its generic name and computing the order of every element. Show all work.

Let A5 denote the alternating group on a 5-element set {1,2,3,4,5}. The set of automorphisms of A5 form a group, denoted Aut(A5). The group of conjugations of A5, denoted Conj(A5), is the subgroup of Aut(A5) consisting of automorphisms of the form γs:=s()s1 where sA5. Explicitly, γs(x)=sxs1 for any xA5.

  1. Prove that the function γ:A5Conj(A5), taking sA5 to γs, is a surjective homomorphism.
  2. Prove that A5 is isomorphic to Conj(A5).

Suppose H is a group of order 15. Prove there does not exist a nontrivial group homomorphism ϕ:D5H, where D5 is the dihedral group with ten elements.


Let S7 denote the symmetric group.

  1. Give an example of two non-conjugate elements of S7 that have the same order.
  2. If gS7 has maximal order, what is the order of g?
  3. Does the element g that you found in part (2) lie in A7? Fully justify your answer.
  4. Determine whether the set {hS7|h|=|g|} is a single conjugacy class in S7, where g is the element you found in part (2).

Let G be the additive group Z2020 and let HG be the subset consisting of those elements with order dividing 20.

  1. Prove H is a subgroup of G.
  2. Find an explicit generator for H and determine its order.

Let G denote the set of invertible 2×2 matrices with values in a field. Prove G is a group by defining a group law, identity element, and verifying the axioms. Credit is based on completeness.


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